McCain said the senators were "ready to negotiate" with their counterparts in the House. Photograph: Shawn Thew/EPA

Two of the US senators behind a major immigration bill said they felt positive about the prospects of reform on Thursday, despite a clear lack of support for the legislation among Republican representatives.

Speaking after a meeting with President Barack Obama, Republican John McCain and Democrat Charles Schumer said they felt "encouraged" that agreement could be reached.

On Wednesday, House Republican leaders criticised what they called the "flawed" legislation, saying they would instead bring about piecemeal changes focused on bolstering border security.

House Republicans may also bring forward a limited proposal to provide a path to citizenship to immigrants who were brought to the US illegally as children.

However after the packed meeting they appeared largely opposed to a central plank of the Senate bill: creating a route to citizenship for an estimated 11 million undocumented workers.

McCain and Schumer seemed undeterred. "If I had to choose a word to describe yesterday's House meeting, it would be 'encouraging'," Schumer said. "Immigration reform has a strong future here in Washington."

He added: "The caucus sent out a message yesterday, which was the right message, which is, doing nothing is not an option."

McCain said the senators were "ready to negotiate" with their counterparts in the House. "We are ready to talk," he said.

The optimistic tone adopted by the senators, among eight who authored the Senate's immigration bill, contrasted with the position adopted by Republican House speaker John Boehner.

"We are not going to do the Senate bill," he said at a press conference. "I don't believe the internal enforcement mechanisms in the Senate bill are going to give the American people the confidence that we will stop this unlimited flow of undocumented workers coming to our country."

He said House Republicans would instead focus on their own, slimmed-down proposals. "Through all the conversations that have occurred, with my own members, with Democrat members, it's clear that dealing with this in bite-sized chunks that members can digest and the American people can digest is the smartest way to go," he said. "And so I'm much more concerned about doing it right than I am in meeting some deadline."

Asked on what Republican rejection of the bill would mean for the party's electoral prospects among Hispanic voters, he replied: "This is more about doing the right thing for the country. If people want to get into a political conversation, I don't think it is appropriate. We've got a broken immigration system. It needs to be fixed."

With Republicans divided, few in Washington expect any significant immigration reform before the August recess.

Obama has so far resisted becoming publicly overly involved in the debate over the immigration bill, amid fears that his intervention could worsen its prospects among conservative Republicans.

His press secretary, Jay Carney, would not be drawn on whether he would now adopt a more proactive role, as some have predicted.

"The president will engage, as he has done all along," he said. "He will make clear his support for, and his insistence upon, comprehensive immigration reform, because it is the right thing for our economy and it is the right thing for our people."

Carney said the White House was pleased that House Republicans accept that something needed to be done about immigration. "We consider that a good sign," he said. "Progress is possible."

"We will acknowledge that there are a lot of obstacles between where we are today and where we want to be, but that has long been the case," he added. "We are hopeful that in the end Congress will act on the will of the American people."

He pointed out that Obama has always made clear that a pathway to citizenship for undocumented immigrants currently in the US was an "essential" part of any reform.

Earlier, McCain suggested it would be a mistake for the president to mount a more public campaign in support of immigration reform as the House takes it up.

"We want to be very careful that we have the president's participation but these members, these Republican House members — many of them are in districts that they will be representing for a long time — do not feel that they have been unduly pressured by the president of the United States," he said. "So I think the president is walking a careful line here, and I think it's the appropriate one."

The meeting of House Republicans on Wednesday, which has underlined the political challenge for advocates of radical immigration reform, was the first such gathering since the Senate approved the bill last month on a bipartisan vote of 68-32.

In a rare intervention, President George W Bush called on Congress to overcome its differences and reach an agreement.

"I don't intend to get involved in the politics or the specifics of policy, but I do hope there's a positive resolution to the debate," he said during a naturalisation ceremony at his presidential library in Dallas, Texas.

"And I hope, during the debate, we keep a benevolent spirit in mind, and we understand the contributions immigrants make to our country."